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## Homework Statement

Which of the following forces is conservative? (a) F = k(x,2y,3z), (b) F=k(y,x,0), and

(c) F=k(-y,x,0). For those which are conservative, find the corresponding potential energy U, and verify by direct substitution that [itex]\vec{F} = - \nabla U[/itex]

## Homework Equations

## The Attempt at a Solution

I have already ascertained which force fields are conservative, and now I am trying to find the potential energy associated with those force fields.

[itex]\vec{F} = - \nabla U[/itex] implies that [itex]F_x \hat{i} + F_y \hat {j} + F_z \hat{k} = \frac{\partial U}{\partial x} \hat{i} + \frac{\partial U}{\partial x} \hat{j} + \frac{\partial U}{\partial x} \hat{k}[/itex].

Using this fact, I set the components equal to equal other, and integrated each individual component. When I went to see if this was the correct answer, I found the answer was a linear combination of the three individual components. Why is that?